The aliens in Independence Day did not kill a majority of the Human population. There were only a few dozen of alien ships. America alone has a dozen+ cities with millions, let alone Europe, China (duh), India (duh), etc etc. Millions died, but nothing that truly set Humanity back. The only things set back were the air-force military branches of the various countries involved. America still has its mighty Navy and Army/Marines!
The aliens did not destroy cities to wipe Humanity out, that was what the armies were for. They destroyed cities to draw in the resistance and wipe it out, or as best they could before the true invasion force came. As for the surviving aliens, there would have been so few, it wouldn't have mattered. Their tech relied largely on the mothership. Their soldiers proved just as weak as Human's, so basically the still largely intact armies of the world would just march in, kick alien ass, and salvage tons and tons of alien tech and be using it inside of twenty years.
Jeff Goldblooms character hacked into the aliens transmissions. Its no different than a hacker getting into things that "supposedly" he shouldn't be able too. Besides, he stumbled into it accidentally, but was smart enough to figure out how to use that to his advantage. Its not beyond the scope of reality, not even by a long shot. If you think about it, base elements and what can be made from them, means the scope of technology is limited (to what degree, who knows), and all forms of technology would have similar principles in how they operate. Some people, like Jeff's char, would be brilliant enough to realize this simple reality.
Independence day is not the best movie to base a fallout type post-alien invasion story off of.